Search: Add API param for specifying the application and timeout

In the internal search api we have a param called `stat_app_name` which is a string we use for tracking stats. It also is used for determining the timeout of the caching layer. By default on the API we have a 2 minute timeout. This is probably too long for some applications (es-wp-query and an es-admin search).

We could add a parameter like that to the api and then have different caching for different applications.

Start by locating the internal search API handling for stat_app_name and the caching-layer timeout. Determine how an application-specific parameter should select timeout and cache behavior, then verify that the default two-minute behavior remains intact and that es-wp-query and es-admin searches can use different settings.

Written by the indexing model from the issue text.